# Class_21 - Introduction to Multiple Regression Analysis You...

• Notes
• 27

This preview shows pages 1–9. Sign up to view the full content.

Introduction to Multiple Regression Analysis

This preview has intentionally blurred sections. Sign up to view the full version.

You will recall that the general linear model used in least squares regression is: Y i = α + bX i + ε I where b is the regression coefficient describing the average change in Y per one unit increase (or decrease) in X, α is the Y-intercept (the point where the line of best fit crosses the y-axis), and ε i is the error term or residual (the difference between the actual value of Y for a given value of X and the value of Y predicted by the regression model for that same value of X). In other words, the regression coefficient describes the influence of X on Y. But how can we tell if this influence is a causal influence?
To answer this question, we need to satisfy the three criteria for labeling X the cause of Y: (1) that there is covariation between X and Y; (2) that X precedes Y in time; and (3) that nothing but X could be the cause of Y.

This preview has intentionally blurred sections. Sign up to view the full version.

(1) We will know that X and Y covary if the regression coefficient is statistically significant . This means that β (the "true" relationship in the universe, i.e., in general) is probably not 0.0. (Remember, b = 0.0 means that X and Y are statistically independent. Therefore, X could not be the cause of Y.) (2) We will know that X precedes Y in time if we have chosen our variables carefully. (There is no statistical test for time order; it must be dealt with through measurement, research design, etc.)
(3) How can we tell if something other than X could be the cause of Y?; as before, by introducing control variables . We saw how this was done with discrete variables used to create zero-order and first-order partial contingency tables. In the case of regression analysis, statistical control is achieved by adding control variables to our (linear) regression model.

This preview has intentionally blurred sections. Sign up to view the full version.

This transforms simple regression into multiple regression analysis . The multiple regression model looks like this: Y i = α + b 1 X 1i + b 2 X 2i + b 3 X 3i + ε i We still have a residual and a Y-intercept. However, by introducing two additional variables in our regression model on the right-hand side (they are sometimes called "right-side" variables), we have changed the relationship from a (sort of zero-order) bivariate X and Y association to a multi-way relationship with two control variables, X 2 and X 3 . We therefore have three regression coefficients, b 1 , b 2 , and b 3 .
The central point is that, when we solve for the values of these constants, the value of b 1 (the coefficient for our presumed cause, X 1 ) now has been automatically adjusted for whatever influence the control variables, X 2 and X 3 , have on Y. This adjustment occurs mathematically in the solution of simultaneous equations involving four unknowns, α , b 1 , b 2 , and b 3 . In other words, instead of describing the gross influence of X 1 on Y as in the simple regression case, in the multiple regression case this coefficient describes the net influence of X 1 on Y, that is, net of the effects of X 2 and X 3 . This is statistical control at its best and is the way we answer the question of non- spuriousness.

This preview has intentionally blurred sections. Sign up to view the full version.

How do we interpret the results?
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern